Mechanisms of Supercooling and Supersaturation

The homogeneous freezing temperature of water is listed at -42C. However, from the equations formulated to find critical radius here, I did not see any factors which restricts the homogeneous nucleation temperature to a certain value. How is the homogeneous nucleation temperature defined? I can see how the probability of a nuclei of critical radius increases with decreasing temperatures. However, in the absence of heterogeneous nucleation sites, would a block of water left at say -20C never freeze, or would it eventually freeze after a very long time? Microscopically how is a solid nucleus formed?

In the same Wikipedia article, I did not understand this statement:

Supercooling brings about supersaturation, the driving force for nucleation. Supersaturation occurs when the pressure in the newly formed solid is less than the liquid pressure

what is meant by solid and liquid pressure, and since they are different, can phase change be considered isobaric?

How is supersaturation achieved? I think one way is to lower the temperature of some water vapour inside an enclosed container with no nucleation sites. Is it possible to generate a supersaturated system by evaporation, such that water continue to evaporate even when P > P_sat(T_system)?

Similar to the above question, if I have liquid water in equilibrium with its vapour (P_vapour = P_sat(T)) inside an closed system with no nucleation sites at temperature T, if I lower the temperature, does the system supersaturate, or does vapour begin condensing?